Stability Domains of the Delay and PID Coefficients for General Time-

Delay Systems

1Elham Almodaresi,

2Mohammad Bozorg,

3Hamid D. Taghirad

1,2Department of Mechanical Engineering, Yazd University, Yazd, Iran

3Department of Systems and Control, Faculty of Electrical and Computer Engineering,

K. N. Toosi University of Technology, Tehran, Iran.

2Corresponding Author : bozorg@yazd.ac.ir

Stability Domains of the Delay and PID Coefficients for General Time-

Delay Systems

Time delays are encountered in many physical systems, and they usually threaten

the stability and performance of closed-loop systems. The problem of

determining all stabilizing PID controllers for systems with perturbed delays is

less investigated in the literature. In this study, the Rekasius substitution is

employed to transform the system parameters to a new space. Then, the singular

frequency (SF) method is revised for the Rekasius transformed system (RTS). A

novel technique is presented to compute the ranges of time delay for which stable

PID controller exists. This stability range cannot be readily computed from the

previous methods. Finally, it is shown that similar to the original SF method,

finite numbers of singular frequencies are sufficient to compute the stable regions

in the space of time delay and controller coefficients.

Keywords: word; PID controller, Rekasius substitution, singular frequency

method, stability boundaries, uncertain delay .

Introduction

Time delays are either intrinsic parts of the system dynamics or are introduced to the

system due to the delays in measurement or actuation. They usually lead to instability

and poor performance and difficulties in tuning of the controller coefficients. Different

tools are developed to compute the stability margins of time delays. Among them are

the linear matrix inequality (LMI) methods (Chen, Gu, & Nett, 1994; K. Gu,

Kharitonov, & J.Chen, 2003; Nguyen, Ishihara, Krishnakumar, & Bakhtiari-Nejad,

2009) and the frequency based methods (Elham Almodaresi & Bozorg, 2009; Keqin

Gu, Niculescu, & Chen, 2005; Sipahi & Olgac, 2005). The LMI methods compute

conservative margins for delays, while the frequency based approaches are less flexible

for more complicated cases of MIMO systems with uncertain models and external

disturbances. For controller design, iterative algorithms are suggested to produce some

delay dependent LMIs in (Du, Lam, & Shu, 2010). Different frequency based

approaches are also taken to define the set of all stabilizing PID controllers for

processes with fixed time delays, namely the D-decomposition method (Saadaoui,

Elmadssia, & Benrejeb, 2008) and the parameter space method (Hohenbichler &

Ackermann, 2003), which both graphically present the set of all stabilizing PID

controllers for processes with fixed time delays. The singular frequency method

(Bajcinca, 2001, 2006; Hohenbichler, 2009), formulates the stability boundary

crossings, as a matrix rank deficiency problem. Another widely-applied approach to

design PID controllers for time delay systems is the extension of the Hermit-Biehler

theorem for quasi-polynomials (Ou, Zhang, & Yu, 2009). This theorem proves the

interlacing property of roots of the real and imaginary parts of a stable polynomial. A

couple of works have also employed the Nyquist plot properties to compute the delay

stability margins for the second-order and all-pole delay systems (Lee, Wang, & Xiang,

2010; Xiang, Wang, Lu, Nguyen, & Lee, 2007). The Lambert W function is used in (Yi,

Nelson, & Ulsoy, 2007) to compute an analytical solution for delay differential

equations. Though, numerical computations are required in some stages of the

algorithm. The delay is assumed fixed. The PI controller is computed for first-ordered

delay systems by (Yi, Nelson, & Ulsoy, 2013), combining the latter method with the

pole-placement technique. An eigenvalue-based method is also employed by (Michiels

& VyhlíDal, 2005; Michiels, Vyhlídal, & Zítek, 2010) to stabilize delay equations with

certain time delays. However, the effect of the small delay perturbations is investigated

on the stability. An uncertainty delay range is considered by (Emami & Watkins, 2009)

to present a graphical method for computing all robust stabilizing controllers. Again, the

delay is assumed a fixed value, the mean of the uncertainty range. Also, the controller

coefficients and a robust performance parameter are swept in this method.

In practice, for many time delayed systems, delays are uncertain but bounded in

defined ranges. For such systems, assuming fixed delays leads to design of fragile

controllers. This is illustrated in (E. Almodaresi & Bozorg, 2014b) by showing the high

sensitivity of the stability of time delay systems to small values of delay perturbation.

Few methods are so far presented to determine the stable regions in the space of the

uncertain delay and the controller coefficients. Among them are the works of (E.

Almodaresi & Bozorg, 2014a, 2014b) which deal with the special cases of first- and

second-order delay systems. The phase conditions of the open loop transfer functions

are employed in (E. Almodaresi & Bozorg, 2014a) to compute the maximum allowable

delay for which some stable gain exists. Also, Nyquist plot properties are used in

(Almodaresi & Bozorg, 2014b) to compute the regions in the plane of the uncertain

delay and the PI controller coefficients, inside which stable gain intervals exist. Both

approaches are applicable just for the first and the second-order plants containing

delays. However, the uncertain delay and controller coefficients are not required to be

swept to plot the stability boundaries in (E. Almodaresi & Bozorg, 2014b).

Time delay is considered a fixed value in previous works when computing the

stabilizing range of PID controller coefficients. In this paper, we aim to consider the

delay as an uncertain parameter and compute the stable domain in the space of − − τ, while calculating the maximum allowable delay for which a stable PID

controller exists. In the next section, the fundamental results of the Singular frequency

(SF) method are presented (Bajcinca, 2001). This method is employed by

(Hohenbichler, 2009) to compute the stable regions in the space of PID coefficients, for

systems with fixed delays. An inequality constraint is used in (Hohenbichler, 2009) as a

stability condition to compute the allowable -intervals. However, if delay is assumed

uncertain, both sides of the inequality will be nonlinearly dependent on the time delay.

This introduces complexities to this problem and requires additional numerical

computations and tackling of convergence problems.

The most important contribution of this paper is to present an analytic technique for

computing the exact delay range in which a stabilizing PID controller exists. This

technique is based on transforming the system equations by the Rekasius substitution to

a new set of equations that can be handled more conveniently. To achieve this, the SF

method is modified and new results are obtained for the transformed system. After

computing the stabilizable delay range, the SF method is used to plot the stability range

in the plane of KI-KD, by sweeping the delay in the computed range. It is also shown

that the singular frequencies of the Rekasius transformed system (RTS), which are

computed from two generator functions, do not interlace before a certain frequency. It

is also proved that the intersections of the singular lines with the - axis, follow a

special trend, which can be used to compute the stability boundaries.

The rest of the paper is organized as follows. In Section 2, the existing SF results for

the time delay systems with fixed delays are explained. Section 3 introduces the

uncertainty in the time delay and extends the results of Section 2 to this case. The

problem of drawing the stability boundaries in the space of the PID controller

coefficients and the uncertain delay, is addressed in this section. First, the framework of

the SF method is revised for the RTS. Then, a technique is presented to compute the

delay stability interval. Singular frequencies, Singular lines and crossing directions

concepts are computed for the RTS. A practical case study is presented in Section 4 to

demonstrate the implementation of the proposed method.

A Singular Frequency Method

Consider the block diagram of a time delay plant

, = ⁄ , (1)

in which, and are polynomials of degrees and , respectively, where

< . Also, = ⁄ + + is a PID controller as in Figure 1, where, ,

and are the controller coefficients. The characteristic equation is

, = + + + , (2 )

where, = .

Figure 1. Block diagram of a time delay system.

Assume = ! +⋯+ ##, # ≠ 0 and = &! +⋯+ &'('( +'. Obviously, &! = 0. This system is retarded, if ) > 2 and is neutral, if ) = 2. The

characteristic equation (2) has an infinite number of roots due to the exponential delay

term. All of these roots must be located in the left half plane (LHP) to ensure the

stability. The controller coefficients where the roots of (2) cross the stability boundary

are the root crossing boundaries. To compute the crossing boundaries, where the roots

of (2) cross the imaginary axis, = ,- is substituted in (2). Then, by decomposing the

characteristic equation into the real and imaginary parts, two equations are obtained

(Hohenbichler, 2009):

− - = .( -/0- −. -12-, (3-a )

= 3.(- 12- +.- /0-4 - = .-⁄ , (3-b )

in which,

.(- = 5-5- + 6-6- 5- + 6-⁄.- = 6-5- − 5-6- 5- + 6-⁄ , (4 )

+ + + , 7 8 − +

where, 5-, 6-, 5- and 6- are the real and imaginary parts of ,- and

,-, respectively. Equation (3-a) is independent of and (3-b) is independent of and . For a defined value of and , Hohenbichler (2009) computes the singular

frequencies -(, -, … by solving .- = from (3-b). .- is called the generator

function and it is shown that the sign of the derivative of this function change

alternately at the singular frequencies, i.e.,

sgn =>?@>@ |@BC ≠ sgn =>?@>@ |@BDEC , 1 = 1,2, …sgn =>?@>@ |@BC = sgn =>?@>@ |@BDGC , 1 = 1,2, … (5 )

The singular lines, which form the root crossing boundaries, are then obtained

in the − plane, by substituting the computed singular frequencies in (3-a). It is

also shown that, although an infinite number of singular frequencies is computed from

(3-b), a finite number of the frequencies need to be considered to compute the singular

lines that form the stability crossing boundaries. This claim is proved by showing that

from a certain frequency onward, singular frequencies interlace with an interval of H/.

Let us call this frequency, the periodicity frequency.

To account for the changes in number of unstable roots of the characteristic

equation on the two sides of a singular line, the concept of root crossing direction is

introduced. This sign change or the crossing direction is investigated in Bajcinca (2001,

2006) and Hohenbichler (2009).

Uncertain Time Delay

In this section, the SF method is extended for the case of systems with uncertain

delays. The main problem caused by assuming delay uncertainty in (3), is that the

singular frequencies and the periodicity frequency will depend nonlinearly on the value

of the delay, resulting in a set of nonlinear equations. To compute the stable regions in

the space of − − τ, first one needs to compute the stable range of delay and to

evaluate the relation between the delay and the periodicity frequency. In this paper, the

Rekasius substitution is employed as an assistant tool to evaluate these. First, the

characteristic equation is put into a matrix form using the Rekasius substitution. Unlike

the SF method, in which one generator function (3-b) is computed, here two generator

functions are derived to compute the singular frequencies for the RTS. The computed

singular frequencies are obtained as functions of the delay. It is shown that the singular

frequencies of the two functions do not necessarily interlace. However, we will prove

that they interlace from a certain frequency (periodicity frequency) onward. Based in

this result, a method is presented to compute the maximum delay for which stable PID

controllers exist. Then, by computing two sets of singular lines from singular

frequencies, the features of the resulting singular lines are investigated. The singular

lines are evaluated in the − plane and the root crossing direction is computed for

each singular line. By sweeping in the stable delay interval, stable regions in the

− − space are calculated.

Singular Frequency Method for RTS

If the exponential delay term in (2) is substituted by a bilinear rational function

of a new parameter J (Rekasius, 1980):

= 1 − J 1 + J⁄ , = ,-, (6 )

in which,

= @ 3K 2(J- + H4, = 1,2, …, (7 )

the characteristic equation of the RTS is obtained as:

M, J = 1 + J + + + 1 − J. (8 )

When (7) holds, the roots of (8) and (2) are identical. Define M,-, J = N-, J +,O-, J. Putting ,-, J = 0 in a matrix equation form, leads to

PN-, JO-, JQ = R-, J S=1 −-0 0 C PQ + P 0-QT + U-, J = V, (9 )

where,

R-, J = P 5- + J-6 J-5- − 6-6- − J-5- 5- + J-6-Q, (10 )

U-, J = P5- − J-6-6- + J-5-Q. (11 )

Since R-, J is a nonsingular matrix, define ℎ-, J and X-, J as

Pℎ-, JX-, JQ = R-, J( PN-, JO-, JQ. (12 )

Then,

ℎ-, J = − - + Y(-, J = 0, (13)

X-, J = - + Y-, J = 0, (14)

Y(-, J = 3R5-1 − J- + 2R6-J-4/3∆[-1 + J-4, (15)

Y-, J = 3−R6-1 − J- + 2R5-J-4/3∆[-1 + J-4, (16 )

R5- = 5-5- + 6-6-, (17)

R6- = 6-5- − 5-6-, (18)

∆\- = 5- + 6-. (19 )

Equations (13) and (14) are functions of - and T, while (3) is a function of -

and . The parameters set, where ℎ-, J = X-, J = 0, is the same set for which

,-, J = 0. For a fixed , the frequencies that satisfy (14) define the singular

frequencies. By substituting these frequencies into (13), singular lines (stability crossing

boundaries) are derived.

Singular Frequencies of Generator Functions

In this subsection, it is shown that the singular frequencies of the RTS system in

(9) are obtained from two generator functions, while these frequencies are computed

from the generator function (3-b) in the SF method. Solving (14) for J, two solutions

are computed:

J(- = ]−R5- + ^_(-` -. _-a , (20 )

J- = ]−R5- − ^_(-` -. _-a (21 )

where,

_(- = ∆\-∆- − -∆\-, (22)

_- = -∆\- + R6-, (23 )

in which, ∆- = |,-| and R5-, R6-, and ∆\- are defined respectively in

(18) and (19). Substituting J(- and J- forJ in (7),

= b(- = @ 3K 2(-. J(- + H4, = 1,2, …, (24)

= b- = @ 3K 2(-. J- + H4, = 1,2, …, (25 )

are respectively computed. By putting the imaginary part of ,-, J in (8) equal to

zero, two generator functions of singular frequencies as b(- = and b- = are

obtained. By solving them, two sets of frequencies are obtained, respectively denoted by

Ω = -(, -… and Ω′ = -(′ , -′ , … . Now, it is shown through an example that the

singular frequencies - ∈ Ω and -′ ∈ Ω′ do not necessarily interlace. The diagrams

b(- and b- are plotted in Figure 2-a, for Y, in (2) such that

:= −0.26791 − 41.6667,: = 279.03 − 2.9781 + 1.(26 )

Figure 2-b shows the singular frequencies for a fixed delay value = 25 as

Ω = 0.0207, 0.0898,0.1033,0.374,0.6266, … , Ω

′ = 0.2598,0.5069, 0.7568,1.007,1.2583,… . (27 )

From Figure 2-b, it is obvious that the singular frequencies do not interlace.

Furthermore, the sign of the derivative of their generator functions do not interlace, i.e.,

sgnnb(- n-⁄ , - ∈ Ω and sgnnb- n-⁄ ,-′ ∈ Ω′ do not change alternately.

This is the counterpart of the interlacing property of the derivative sign of the generator

function .- in the SF method, which was shown in (5). Although the frequencies of

the sets Ω and Ω′ do not necessarily interlace, they interlace from a certain frequency

onward with the a periodicity of H/. This concept which is shown via the curve -o on Figure 2, will be discussed later.

Definition 1. Define

= S + 2 2⁄ if + 2even + 2 − 1 2⁄ if + 2oddw. (28)

where, and 2 are respectively the degrees of and . Also, define the

following conditions:

C.1: # > 0&evenor # < 0&odd, (29)

C.2: # > 0&oddor # < 0&even, (30)

where, # is the coefficient of # in .

Proposition 1. For a large enough value z ∈ ℕ, the singular frequencies -| ∈ Ω and

-|′ ∈ Ω′, , = z, z + 1, . . . interlace with the periodicity of H/.

Proof. By substituting (20) and (21) respectively into (24) and (25), the following

equations are respectively obtained:

- = cos-/2 ~^_(- − R5- − sin-/2 _- = 0, (31)

′- = cos-/2 ~^_(-+R5- + sin-/2 _- = 0 (32)

Figure 2. a) The diagrams b(- and b- for = 0,⋯ ,5. b) Frequency roots of

b(- = 25 and b- = 25. The curve -o is introduced in the subsection “Stable

Delay Interval”.

0 0.2 0.4 0.6 0.80

50

100

ω

τ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

24

24.5

25

25.5

ω

τ

f1(ω)

b)

a)

ω'

1

* ** * * * *ω

3

ω'

3ω

5

ω'

2ω

4

ω2

ω1

*

f2(ω)

ω(τ)

The roots of - and ′- respectively define the singular frequency sets Ω

and Ω′. Obviously, these sets depend on the delay value. Denote the roots of the

dominant terms of - and ′- by -∞ and -′∞. It can be easily checked that, for

high frequencies, the root chains of - and ′- lie on the roots of sin-/2 = 0

and cos-/2 = 0, respectively, if and 2 are even numbers and Condition C.1 in

(29) is satisfied, or if and 2 are odd numbers and C.2 holds. Hence,

-∞ = 2H/, -′∞ = 2H/ + H/. (33)

For the case that and 2 are even numbers and C.2 holds or and 2 are odd numbers

and C.1 holds, -∞ = 2H/ + H/ and -′∞ = 2H/ are obtained. The root chains of

- and ′- ultimately lie on the roots of

tan- 2⁄ = −1, (34)

tan- 2⁄ = 1, (35)

respectively, if and 2 are respectively even and odd numbers and C.1 holds or if

and 2 are respectively odd and even numbers and C.2 holds. In this case, -∞ =2H/ − H/2 and -′∞ = 2H/ + H/2 hold. For the case that and 2 are

respectively even and odd numbers and C.2 holds or and 2 are respectively odd and

even numbers and C.1 holds, -∞ = H/ + H/2 and -′∞ = 2H/ − H/2 are

obtained.

For all above cases, -∞ − -′∞ = H/ holds. Therefore, a large enough value z ∈ ℕ

exists such that

-| − -|′ − H/ ≤ /, ≪ 1,, = z, z + 1, …, (36)

where , is an integer number. Obviously, z depends on the small value . Hence, the

proposition is proved.

The above proof indicates that the root chains of - and ′- ultimately lie on the

roots of two sinusoidal functions with a phase difference of H/2. Hence, z is a finite

number depending on .

Remark 1. Note that the indices of -| and -,+, ′ in (36) are not necessarily identical. For

instance, the singular frequencies (27) tend to the periodicity of H/ with precision

= 0.2 for the sequence -M , -, -M , -, -M , …, which are the frequencies -| ∈ Ω,

-|′ ∈ Ω′ for , ≥ 5, (Figure 2).

Definition 2. The high singular frequencies are the frequencies -| ∈ Ω and -|′ ∈ Ω′,

, = z, z + 1, . . ., ∈ ℤ, such that (36) holds. Also, define the periodicity frequency

-o = max-, -′ , (37)

such that the singular frequencies larger than that, interlace with the periodicity of H/.

Using the above results, a method is presented in the next subsection to compute the

stable delay interval.

Stable Delay Intervals

In this subsection, a new method is presented to compute the delay interval for which

stabilizing PID controllers exist. The singular frequency sets Ω and Ω′, computed

respectively from (24) and (25), change when the delay is changed. From Definition 2,

all the singular frequencies greater than -o almost reach the periodicity H/. Obviously,

-o depends on the delay. Define as the number of singular frequencies in the

interval 0, -o. A necessary condition for the stability of (2), is that (Hohenbichler,

2009)

≥ #6', (38)

#6' = ceil =####,# + @o. C − 1 (39)

in which, and are respectively, the number of unstable roots and the roots

located at the origin, of . is the number of roots = ,-| of with -| ≠ 0.

of the roots of posses an odd order and ensure an existing limit

lim@→@ |Y,-, ,-⁄ |. and Y, are respectively defined in (1) and (2).

Also, rem, 2 denotes the remainder of the division of by 2.

Although, (39) is a necessary condition for the stability, it can be used to find a

stable sweeping interval when delay is assumed fixed. However, assuming the delay

to be uncertain, both sides of the inequality depend on the delay value. Therefore, -o must be evaluated first to compute #6'. Here, a technique is presented to find -o.

From (36), there exists a large enough value z ∈ ℕ, for each known delay , such

that - − -′ equals to H + ⁄ or H − ⁄ , ≪ 1 (Proposition 1). Define

-¡′ = -′ for simplicity. Without loss of generality, assume

- − -¡′ = H − ⁄ . (40)

Since - and -¡′ are the roots of (24) and (25), respectively, one has:

= 2 3K 2(-. J(- + H4 -⁄ , (41)

= 2 ]K 2(-¡′ . J-¡′ + H` -¡′a , (42 )

where, is an integer number. By substituting from (40) into (41) and (42)

-J(- = /( + /-¡′ 1 − /(/-¡′ a , (43 )

is obtained, where /(: = tanH − 2⁄ and /-¡′ = -′J-¡′ . By substituting - into (20),

-J(- = −R5- + ^_(- _-a (44)

holds. Define the right side of (43) as / ′-¡′ . Then, (43) simplifies to

R5- + / ′-¡′ _- = ^_(-, (45)

by using (44). For a fixed -¡′ , (45) is a nonlinear function of -. Square two sides of

(45) to obtain the linear equation

]R5- + / ′-¡′ _-` = _(-. (46)

Then, - is computed from the polynomial function (46) by sweeping -¡′ . Obviously,

those pairs of -,-¡′ are acceptable which satisfy (45). For each pair of -,-¡′ , is

evaluated from (40). Then, -o is computed from (37). Now, #6' is calculated by

substituting -o into (39). Also, is the number of singular frequencies in the

interval 0, -o. Finally, the stable delays are the delays satisfying (38). The -o diagram is plotted in Figure 2 for the system of (26). More explanations are provided at

the case study section.

Singular Lines and Root Crossing Direction

In this subsection, the singular lines and the root crossing directions are

investigated for the RTS (8).

Definition 3. Define two sets of singular lines ¢ = ℓ(, ℓ, … and ¢′ = ℓ(′ , ℓ′ , … computed respectively by substituting the singular frequencies of the sets Ω and Ω′

into

(13). Also, define the -intersection, as the intersection of a singular line with the axis, denoted by ∗.

Remark 2. The more stable side of a singular line is defined as its side with less closed-

loop poles which is indicated as the unmarked side in all graphs of this paper.

Lemma 1. The -intersections of the singular lines ℓ| ∈ ¢ and ℓ|′ ∈ ¢′, , = 1,2, … are

respectively negative and positive.

Proof. From (15), Y(-, J is rewritten as

Y(-, J = ∆@¤¥G@G∆¦@§¨@©@§B@©@G¤¥∆¦@ , (47)

by using (14) and (16). Also, ∗ = −Y(-, J is obtained from (13). Substitute J( and

J respectively from (20) and (21), for J in (47). Then,

∗ = ª−b-|,-| ∈ Ωb-|′ ,-|′ ∈ Ω′w,b- = «∆¬¤¥G∆¦@G

∆¦ . (48 )

Assume b- ≠ 0, ∀-. Since b- is positive for all frequencies, the lemma is proved

(Figure 3).

Figure 3. Singular lines and -intersections on the − plane.

Corollary 1. From Lemma 1, singular lines ℓ| ∈ ¢ and ℓ|′ ∈ ¢′, , = 1,2, … are

respectively computed from

ℓ,′ ∈ ¢′

° °∗ ∗ ℓ, ∈ ¢ ∗ ∗ ∗ ∗

ℓ|: − -| + b-| = 0, -| ∈ Ω, (49 )

ℓ|′ : − -|′ − b-|′ = 0, -|′ ∈ Ω′ . (50 )

In the rest of this subsection, the root crossing direction is computed for the

singular lines (49) and (50). Assume the singular line in Figure 4 is crossed in the

direction ± > 0. The direction of crossing of eigenvalues of the system is defined by a

root crossing index (Bajcinca, 2006)

= sgn3Re³ ³⁄ 4|´|@. (51)

Figure 4. The root crossing direction for a singular line ℓ.

Suppose that the singular line is crossed in the direction ± > 0. The system’s

eigenvalues cross the imaginary axis towards the RHP if = 1 and towards the LHP if

= −1. The following theorem computes the root crossing direction of the singular

lines of the RTS (8).

Theorem 1. The root crossing index for the singular lines (49) and (50) is evaluated

from

= −sgn>µ@,©>@ + >µ@,©>© >©@>@ . (52 )

where,X-, J is defined in (14), and J- = 3tan- 2⁄ 4/- is computable from (7).

Proof. Since and are fixed, , J is a function of , J and from (8).

ℓ n°> 0°

Furthermore, since , J = 0, the total differential equation ³, J is equal to zero

(Bajcinca, 2006). Hence,

³ = >¶> ³ + >¶>© >©> ³ + >¶>¤ ³ = 0. (53 )

From (53),

··¤ = − >¶>¤ ~>¶> + >¶>© >©>a , (54 )

holds. By substituting ,-, J = O-, J + ,N-, J and = ,- into (54), is

computed from (51) and (54) as

= sgn ¸>¹@,©>¤ >¹@,©>@ + >¹@,©>© >©>@>º@,©>¤ >º@,©>@ + >º@,©>© >©>@¸. (55)

Then, (55) is simplified to

= sgn =−|R-, J| ~>µ@,©>@ + >µ@,©>© >©>@C, (56 )

by using the relationship (12) between N-, J, O-, J and ℎ-, J, X-, J, where |. | is the determinant function. From (10), |R-, J| = ∆\-1 + J- holds. Since

∆\- in (19) is non-negative for all frequencies, (52) holds.

It is proved in (Hohenbichler, 2009) that for each certain delay, only the singular

lines corresponding to the singular frequencies smaller than the periodicity frequency

are needed to be computed. It is also shown that the stable region which is a polygon for

retarded systems can be described by the limit of a sequence of polygons for neutral

systems. Detailed proof can be found in (Hohenbichler, 2009).

Remark 3. It can be shown that the singular lines of the RTS, corresponding to the

frequencies higher than the periodicity frequency have some special features. The more

stable side of the singular lines ℓ|, , = z, z + 1,… points to the side where increases.

On the other hand, all singular lines of the set ¢ intersect with the negative -axis

(Lemma 1). Therefore, the more stable side of ℓ| ∈ ¢, , = z, z + 1, … always points to

the origin (Figure 5). On the other hand, the junctions of singular lines ℓ|, , = z, z +1, … move increasingly to the right half-plane for a retarded system (Figure 5) and tend

to a positive value for a neutral system (Hohenbichler, 2009). Therefore, the

junctions »| ∈ ¼ move outward from the stable polygon for , = z, z + 1,…. Similar

results are deduced for singular lines ℓ|′ , , = z, z + 1,…. Hence, the finite set of singular

lines ℓ| ∈ ¢, ℓ|′ ∈ ¢′, , = 1,2, … , z − 1 are sufficient to be used to compute the stable

regions in the − plane for each .

A similar concept of periodicity frequency is also used in Hermite-Biehler

technique of Ou, Zhang, & Yu (2009). The roots of the real and the imaginary parts of

the characteristic equation interlace for the frequencies larger than this frequency.

Hence, just the frequencies smaller than the periodicity frequency are required to be

computed.

Figure 5. The singular lines and their intersections.

Remark 4. The real root boundary in − plane is the line = 0 and its more

stable side is described by

ℓ, ℓ,+1 ℓ,+2

», »,+1

°

> 0, if > −&( !⁄ , ! ≠ 0, (57 )

< 0, if < −&( !⁄ , ! ≠ 0. (58 )

The proofs of (57) and (58) is straight forward by setting the frequency in (13) and (14)

equal to zero to obtain the RRB and using (51) to compute the root crossing direction.

Algorithm

1. Computing the stable delay interval:

1.1. Choose a small enough ≪ 1. Sweep -¡′ ∈ 30,∞ to compute - from the

polynomial function (46). Select the pairs - and -¡′ that satisfy (45).

1.2. Evaluate from (40) for each pair - and -¡′ , and compute -o from (37).

1.3. Calculate #6', by substituting -o into (39), and as the number of

singular frequencies in the interval 0, -o. Then, compare the diagrams

and #6' and find the delay interval for which is greater than

#6'. This is the stable delay interval, inside which (38) is satisfied.

2. Plotting the stable regions in the − − space:

Iterate on to sweep the interval computed in Step 1, and do the following in each

iteration:

2.1. Find the value of z so that (36) holds. Then, compute the singular frequencies

-| and -|M, , = 1,2, … , z − 1 from (24) and (25).

2.2. Compute the singular lines ℓ| and ℓ|M, , = 1,2, … , z − 1 by substituting the

singular frequencies of step 4 into (49) and (50). Then, Compute corresponding to each singular line from (52). Mark the upper or lower side of

a singular line if = 1 or = −1 hold, respectively.

2.3. Mark the upper or lower side of the RRB ( = 0), respectively if (58) or (57)

holds.

2.4. Define the stable regions as the regions surrounded by the unmarked sides of

singular lines and the RRB

Case Study

Consider the second-order, unstable, non-minimum phase time delay plant of a

stirred tank reactor (Rao & Chidambaram, 2006) in (26). A PID controller along with a

lead/lag controller structure is tuned to stabilize the plant (26) with a fixed delay of 20s.

Here, the controller is a PID controller with = 1 (Figure 1). The algorithm of

Section 4 is used to compute the stable domain in the − − space.

Step 1.1: A small value of = 0.2 is chosen. Then, -¡M in (46) is swept in ∈ 30,∞ to

compute -. The -, -¡′ pairs are chosen to satisfy (45). For instance, - = 0.493 is

computed for -¡M = 0.376.

Step 1.2: For each computed pair, is evaluated from (40). For instance, =25

corresponds to -¡M , - = 0.376,0.493. -o corresponding to each , equals to

max-, -¡M from (37). In this case, -o25 = max0.376,0.493 = 0.493. Figure 2

shows the -o diagram. Note that -o is on the horizontal axis. The number of

frequencies at the left side of this curve is #6'. As stated in Remark 1, the sequence

-M , -, -M , -, -M , … have the periodicity H/ with precision = 0.2 for = 25. These

frequencies are at the left side of -o25. Step 1.3: The values ) = 2, = 1 and = = = 0 are computed using (26).

By substituting these values and -o in (38), #6' = ceil33 2⁄ + -o H⁄ 4 − 1 is

computed. For instance, #6'25 = ceil33 2⁄ + 0.493 ∗ 25 H⁄ 4 − 1 = 5 holds. Also,

25 = 5 is obvious from Figure 2. #6' and are plotted in Figure 6.

Obviously, exceeds #6' in the interval ∈ 30,25.454, satisfying (38). Hence,

= 25.45 is the maximum delay for which stable PID controller exists for = 1. It

can easily be checked that this maximum can be increased to = 33.35 by decreasing

to 0.4. The computed interval is swept to perform Step 2. Consider = 25 ∈30,25.454. The periodicity of the high singular frequencies of (26) for = 25 is

already investigated in Remark 1.

Step 2.1: from Remark 1, the singular frequencies -, -… ∈ Ω and ½-′ , -′ , … ¾ ∈Ω

′ in (27) tend to the periodicity H/ with precision = 0.2. Hence, from (27), the

singular frequencies greater than 0.374 in the Ω set and greater than 0.2598 in the Ω′

set, interlace. Hence, the singular frequencies 0.0207, 0.0898, 0.1033, 0.374 in the Ω

set, and 0.2598 in the Ω′ set, are sufficient to be checked to compute the stable

regions.

Step 2.2: The corresponding singular lines ℓ(, … , ℓ and ℓ(′ are computed in the plane of

− , by substituting the singular frequencies -(, … , - ∈ Ω and ½-(′ ¾ ∈ Ω′,

respectively into (49) and (50) (Figure 7). For instance, the singular frequency ℓ( is

computed as ℓ(: − 0.0004 + 0.0473 = 0 by substituting -( = 0.0207 in (49).

Also, the more stable side is evaluated from (52). For instance, = −1 is computed by

substituting -( = 0.0207 in (52).

Step 2.3: The RRB is the line = 0, and its marked side is computed by checking (57)

and (58).

Step 2.4: The stable region surrounded by the unmarked sides is shown in Figure 7. By

sweeping in the stable interval 30,25.454, the stable region is obtained in the 3D space

of − − (Figure 8).

Figure 6. #6' and .

Figure 7. Stable region in − plane for = 25.

0 10 20 30 40 50 60

2

3

4

5

6

7

8

9

10

τ

z

zmin

25.45

*

0 5 10 15 20 25 30

-0.04

-0.02

0

kD

kI

Stable region

RRB

l3

l'

2

l'

1

l2

l1

l4

Figure 8. Stable region in the − − space.

Conclusion

Several existing tools are used in this paper to compute an unsolved problem for

PID control of time delay systems with perturbed delays. A general class of time delay

systems with arbitrary orders is considered. The characteristic equations of the system

are transformed to a new space using, the Rekasius substitution. The singular frequency

method is reframed here for the Rekasius transformed system (RTS). It is shown that, in

this case, two generator functions are obtained for computing two sets of singular

frequencies. Also, it is proved that the frequencies of two sets interlace with a

periodicity of to the periodicity of H/ at high frequencies. Then, two sets of singular

line and their features are extracted from the RTS equations. The singular frequencies

and periodicity frequency are computed as functions of the uncertain delay. Then, the

delay intervals for which stable PID controllers exist, are computed. An algorithm is

presented to determine the maximum allowable time delay, for which a stabilizing PID

controller can be found. To the best of authors’ knowledge, no analytic method is

available in the literature, to compute this delay. All the previous works have assumed

-200

20

-0.06

-0.04

-0.02

0

5

10

15

20

25

τ

KI

KD

fixed delays to compute the set of all stabilizing controllers. Even when uncertain delay

is assumed as in Emami & Watkins (2009), a mean value in the range of the uncertainty

delay is assumed. This leads to rough estimation of stabilizing controllers. Finally, the

stability regions in the space of delay and controller coefficients are computed by

sweeping the delay in the computed stable range. The extension of the results to

compute the range of delay and the controller coefficients, to satisfy performance

specifications such as reference input tracking and disturbance rejection, is an open area

for further research.

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